Let \(C\subsetneq X\)  be a proper closed subset and let \(Y_i\)  be integral \(K\) -varieties, endowed with finite surjective separable morphisms \(\pi _i \colon Y_i \rightarrow X\)  of degree at least two, \(i= 1,\ldots ,n\)  (cf. Lemma REF ). We have to prove that \(\Gamma \lnot \subseteq C \cup \bigcup _{i=1}^n\pi _i(Y_i(K))\) .
Since the extensions of function fields induced by \(\pi _i\)  are separable, there exists a dense open subscheme \(W\subseteq X\)  such that, for every \(i=1,\ldots ,n\) ,
the restriction \(\pi _i^{-1}(W) \rightarrow W\)  is finite étale.
By this and the openness of the normal locus [1]}, we may choose a normal dense open subscheme \(S^{\prime }\subseteq S\)  and a normal dense open subscheme
\(X^{\prime } \subseteq f^{-1}(S^{\prime }) \setminus C \subseteq X\) 
such that \(f\)  restricts to a finite étale morphism of normal \(K\) -varieties \(f^{\prime } \colon X^{\prime }\rightarrow S^{\prime }\) .
Set \(Y_i^{\prime } = \pi _i^{-1}(X^{\prime })\subseteq ~Y_i\)  and denote the restriction of \(\pi _i\)  by \(\pi _i^{\prime } \colon Y_i^{\prime } \rightarrow X^{\prime }\) . Then \(\pi _i^{\prime }\)  is dominant and generically finite of the same degree as \(\pi _i\) .
Furthermore, \(\Sigma \cap S^{\prime }\)  is not thin in \(S^{\prime }\)  and, for \(s\in \Sigma \cap S^{\prime }\) , the fibre \(X^{\prime }_s\)  is an open subset of \(X_s\) , so that it is integral (by our assumption that \(X_s\)  is integral) and \(X^{\prime }_s \cap \Gamma \)  is not thin in \(X_s\) . By Lemma REF , this gives us
\(\Gamma \cap X^{\prime } \lnot \subseteq \bigcup _{i=1}^n \pi _i^{\prime }(Y_i^{\prime }(K)).\) 
Therefore,
\((\Gamma \cap X^{\prime }) \setminus \bigcup _{i=1}^n \pi _i^{\prime }(Y_i^{\prime }(K)) \ne \emptyset \) . Obviously,
\((\Gamma \cap X^{\prime }) \setminus \bigcup _{i=1}^n \pi _i^{\prime }(Y_i^{\prime }(K)) = (\Gamma \cap X^{\prime }) \setminus \bigcup _{i=1}^n \pi _i(Y_i(K)).\) 
Finally, by our definition of \(X^{\prime }\) ,
\((\Gamma \cap X^{\prime }) \setminus \bigcup _{i=1}^n \pi _i(Y_i(K)) \subseteq \Gamma \setminus \left( C \cup \bigcup _{i=1}^n \pi _i^{\prime }(Y_i^{\prime }(K)) \right).\) 
We conclude that \(\Gamma \lnot \subseteq C \cup \bigcup _{i=1}^n\pi _i(Y_i(k))\) , as required.
